Measurement of electronic transport properties
3.3 The Hall effect and semiconductor statistics
In this section we review elements of semiconductor statistics that are of direct relevance to work in this thesis. The reader should refer to reference [114] for more detail.
Chapter 3: Measurement of electronic transport properties 43
3.3.1 Concentration of electrons in the conduction band
In general, we are concerned with calculating the electron concentration in the conduction band of silicon. In quite general terms, we can write this down as the product between the probability f (E) that an electron occupies a state of energy E with the density of states g(E), integrated over the energy of all states in the conduction band:
n = Z ∞
Ec
f (E) · g(E) · dE, (3.13)
where Ec is the bottom of the conduction band, and we have set the upper limit of in-tegration to ∞. Strictly, we should integrate only over the energy range covered by the conduction band of silicon, but the upper limit of integration is unimportant provided it is above any occupied state. The integrand of equation 3.13 is composed of the occupation probability, f (E), which has the form of the well-known Fermi function
f (E) = 1
1 + exp(E−Ek F
bT ), (3.14)
and the density of states g(E), which we approximate near the bottom of the conduction band in silicon with the form
g(E) = 4π 2m∗e h2
3/2
(E − Ec)1/2. (3.15)
where m∗e is the effective mass of an electron near the conduction band minimum, h is Planck’s constant, and Ec and EF are the positions of the conduction band edge and Fermi level, repsectively. Performing some elementary operations, we can write the integral of 3.13 as
Chapter 3: Measurement of electronic transport properties 44
In equation 3.16 we have introduced the so-called reduced Fermi energy η, a dimensionless quantity that expresses the distance of the Fermi level from the conduction band edge as a multiple of the characteristic thermal energy kbT :
η = EF − Ec
kbT . (3.17)
We emphasize that η < 0 when the Fermi level is below the conduction band, and thus within the band gap. Also, we can rewrite equation 3.16 in a simple form by grouping the terms
F1/2(η) = 2π−1/2 Z ∞
0
1/2· d
1 + exp( − η), (3.18)
which is known as the Fermi-Dirac integral, and takes as an argument the reduced Fermi energy η; and
Nc(T ) = 2 2πm∗ekbT h2
3/2
= 5.45T3/2× 1015cm−3· K−3/2, (3.19) which is known as the band edge density of states, and has a value of about 3 × 1019cm−3in silicon at room temperature. Using the substitutions of equation 3.18 and 3.19, the carrier concentration in the conduction band take the following simple form:
n(T, η) = Nc(T )F1/2(η). (3.20)
In general, the Fermi-Dirac integral is one of a class of integrals Fj(η) defined by the subscript j. We will only have need of the j = 1/2 member of this family, and will drop the subscript for the remainder of this thesis. Further, we separate our analysis into systems that are non-degenerate, loosely defined as systems for which the dimensionless quantity η is several integers less than zero, and systems that are degenerate, which corresponds to any
Chapter 3: Measurement of electronic transport properties 45
system where this condition is not met. The term degenerate comes up frequently in the analysis of the carrier concentration, and in the context of semiconductor statistics, it simply refers to any instance in which η is close to or larger than 0. Historically, the term comes from the fact that the heat capacity of metals, governed by the limited number of electrons near the Fermi level that can participate in the absorption of heat, is much smaller than would be predicted classically. This phenomena was referred to as the “degeneration” of the heat capacity to small values. Thus, any context in which the Fermi level is near or within a partially filled band − and thus reduces the number of electrons that can participate in a particular phenomena relative to classical expectations − is considered degenerate.
However, returning to semiconductor statistics: in practice, the non-degenerate limit obtains when η is less than about −2, in which case we do not lose much accuracy if we approximateF (η) = exp(η). In this case, we can write our original equation for the carrier concentration simply as
n(T, η) = Nc(T )exp(η). (3.21)
Experimentally, we will often need to test for the condition of non-degeneracy when we have measured the carrier concentration over a particular temperature range. Referring to equation 3.20, we see that this amounts to determining whether the relationship
n(T )
Nc(T ) ≤F (η = −2) ≈ 0.13, (3.22)
is maintained over our measurement range. Such a determination is calmingly simple to make.
Chapter 3: Measurement of electronic transport properties 46
compensating states donor states conduction band
valence band density NA density ND
un-occupied state EC
Energy
EC–Ed
EV
occupied state
Figure 3.2: A schematic model of a donor-dominated semiconductor at T = 0 K. In this example, donors are characterized by a density Nd and a binding energy Ed. A much lower concentration Naof compensating impurities introduce hole-like states to which donor electrons relax in all temperature ranges of interest.
3.3.2 Impurities and their excitations in a silicon lattice
Dopants are often added to silicon to alter its electrical and optical properties.
Dopants that introduce excess electron states into the band gap, which through thermal ionization can be excited into the conduction band, are called donors. Alternatively, dopants that introduce hole-like states into the band gap, which through thermal excitation can be excited into the valence band, are called acceptors. When a donor (acceptor) state is occupied by an electron (hole), we consider it neutral; otherwise we consider it ionized. In addition to their density Nd, donor states are characterized by the energy Ed required to remove an electron from a neutral donor atom and place it in the conduction band with energy Ec. Acceptors are likewise characterized by their density Na and the energy Ea required to remove a hole from a neutral acceptor atom and place it into the valence band with energy Ev.
Chapter 3: Measurement of electronic transport properties 47
In this thesis, we are generally interested in the case where Na Nd. In this case, even at T = 0 K, a portion of the donor state electrons will relax to the lower energy states available in the holes introduced by the acceptors, and a density Na donor states will be empty. The ionization of donor states in this fashion is known as compensation. In this case, Eais unimportant (indeed, compensation generally arises from a variety of states throughout the band gap), and the system is characterized by Na, Nd, and Ed. Additionally, at finite temperatures, a portion of the remaining donors will be ionized through thermal excitation of the electrons to the conduction band.
The total occupation of the donor states can be expressed statistically as a function of the Fermi level EF and the temperature T , but we must consider this problem carefully.
In general, donor states have a level of spin degeneracy for which we must account. For example, a simple hydrogen-like donor can, in its ground state, host an electron of either spin up or spin down. We describe such a donor as having spin degeneracy β = 2. However, β can take on a variety of values depending of the details of the dopant, but will in general be on order 1. For a general donor state of spin degeneracy β, the ionized fraction will be:
Nd+ = Nd
1 + β−1exp
hEF − EC + Ed
kbT
i
= Nd1 + β−1exp (d+ η)−1
, (3.23)
which can be derived from the statistical mechanics of fermions, and where we have re-introduced the reduced Fermi energy of equation 3.17, as well as the reduced binding energy
d= Ed/kbT .
Above, we derived equation 3.20 to be the the electron concentration in the con-duction band. This equation is true regardless of whether we have doped a semiconductor or not, provided the approximation of a parabolic conduction band minimum is valid. For
Chapter 3: Measurement of electronic transport properties 48
a doped, compensated semiconductor, however, we can constrain the concentration of free electrons in the conduction band n to be equal to the concentration of ionized donors, minus the concentration of compensating impurities:
n(T, η) = Nc(T )F (η)
= Nd+− Na
= Nd1 + β−1exp (d+ η)−1
− Na. (3.24)
There is a unique value of η that satisfies this equation for a particular value of T , Na, Nd, Ed, and β. In general, this form of equation is not useful for us experimentally, though, as both T and η vary with temperature, and we do not immediately have knowledge of η. Without knowing η, we cannot fit equation 3.24 for the temperature independent (and material specific) quantities Nd, Na, Ed, and β. We would prefer an equation of the form:
n ≡ n(T ; Nd, Na, Ed, β), (3.25) such that we can treat the temperature independent quantities as fitting parameters, and provided we have at least four (n, T ) data points, solve for them. Generally we will collect many more data points, so as to over-constrain the problem. In the next section we describe how this is done.
3.3.3 Fitting for the energy of a donor electron state
Good approximations exist for F (η) even in the slightly degenerate regime: for η ≤ +1), we can use approximations of the form
F (η) = [C + exp(−η)]−1. (3.26)
Chapter 3: Measurement of electronic transport properties 49
As we mentioned in equation 3.21, if η ≤ −2, we can set C = 0 and F (η) ≈ exp(η) is an accurate approximation. For larger values of η, but still η ≤ 1 we can use C = 0.27 without introducing an error of more than a percent or so [114]. Substituting 3.26 into equation 3.24, we can rearrange it as a quadratic in exp(η). Solving this equation yields the following solution for the Fermi level:
By simply inserting this equation into equation 3.20 we obtain for the carrier concentration:
n = 2Nc(Nd− Na)
As we have previously discussed, when the system is comfortably non-degenerate (η ≤ −2), we set C = 0 and equation 3.28 becomes
n = 2(Nd− Na)
Chapter 3: Measurement of electronic transport properties 50
and the Fermi level is given simply using equation 3.21:
Ec− EF = kbT exp(n/Nc). (3.30)
The equations 3.28 and 3.29 are precisely of the form we sought (n ≡ n(T ; Nd, Na, Ed, β)).
The only temperature-dependent parameter on the right hand side is the temperature itself, which enters through NC and d = Ed/kbT . Thus, given a set of measurements of n as a function of T , we can find the best fit values for Nd, Na, Ed, and β in a least squares sense using any numerical technique that suits us.
3.3.4 Developing statistical intuition
Limiting behaviors of the carrier concentration
There are several limiting cases and simple examples that we will be well-served to consider. First, we consider the behavior of the carrier concentration as a function of temperature, via equation 3.29. We briefly consider three limits. First, when d → 0 (kbT Ed), the denominator is equal to 2, and the carrier concentration becomes
n(T ) = Nd− Na, (3.31)
which represents the expected result that, at high enough temperatures, all donors are ionized; the carrier concentration is thus equal to the donor concentration minus the con-centration of any compensating impurities. As we cool the sample down, n will eventually begin to decrease as donor states begin to freeze out. If the degree of compensation is small, such that it is possible for Na n Nd, we can approximate equation 3.29 as:
n(T ) ≈p
βNcNdexp
− Ed 2kbT
. (3.32)
Chapter 3: Measurement of electronic transport properties 51
In this regime, a plot of log(n) versus 1/T will exhibit a slope of Ed/2. We refer to this temperature regime, if the degree of compensation is small enough for it to exist, as the donor -dominated freeze-out regime. If we continue cooling, we will eventually reach the compensation-dominated freeze-out regime, for which — when we examine 3.29 in the limit that n Na Nd — we can write:
In the compensation-dominate freeze-out regime, a plot of log(n) versus 1/T will exhibit a slope of Ed. We have shown these regimes schematically in Figure 3.3. Provided we are studying a sample in non-degenerate conditions, consideration of these limits often gives us a starting point for a fit or insight into the value of the binding energy Ed.
1 / T
Figure 3.3: Typical ionization of a monovalent impurity in silicon. In addition to the regimes discussed in the text, we also show the intrinsic regime, in which excitation of electrons from the valence band to the conduction band dominates the behavior of n(T ). This regime has a significantly steeper slope in this plot, equal to half the band gap of the material.
Chapter 3: Measurement of electronic transport properties 52
Behavior of the Fermi level
We will often find it useful to speculate intelligently on the effect that changing the value of Ndand Edhas on the position of the Fermi level EF. In an extrinsic semiconductor, the Fermi level is fixed by the constraint of equation 3.24. If we consider only cases that are not more than mildly degenerate (η ≤ 1), we can make use of the approximation of equation 3.26 with C = 0.27, and arrange equation 3.24 in a dimensionless form:
1
0.27 + exp(−η) = Nd/Nc
1 + β−1exp (d+ η) −Na
Nc. (3.34)
For clarity we have not written explicitly Nc’s dependence on temperature, and we remind the reader that η = (EF − Ec)/kbT and d = Ed/kbT . We first consider the form of both sides, noting that the left-hand side (LHS) is a monotonically increasing function of EF, while the right-hand side (RHS) is a monotonically decreasing function of EF. This observation indicates, provided Nd, Ed, and Na take experimentally reasonable values, the two functions will have to intersect at some value of EF. The value of EF at intersection is then the value that satisfies equation 3.34, and the value of EF in our system.
We can consider the solution of this equation graphically, plotting both the left-and right-hleft-and sides as a function of the value of the Fermi level EF. The value for which the two curves intersect is the solution to the equation, and the value of the Fermi level in our system. We consider a system at T = 300 K, with a compensation fraction Na/Nd= .001, a value representative of most of the samples we will study in this thesis.
In the top portion of Figure 3.4, we see the effect that changing the concentration of donors (Nd) has on our system. Because Nddoes not effect the fraction of ionized donors at a particular value of temperature and Fermi level, a change of Ndsimply shifts vertically the curve reflecting the RHS of equation 3.34. This shift forces the the Fermi level that
Chapter 3: Measurement of electronic transport properties 53
Figure 3.4: Graphical techniques for finding the Fermi level. The value of EF corresponds to the solution of equation 3.34, which we find graphically as the point of intersection between the two sides of the equation. For this plot, we use T = 300 K, β = 2, and Na/Nd= 10−3; at this temperature Nc= 3 × 1019cm−3. Top: Decreasing the doping concentration Ndwill always shift the location of intersection (and thus the value of EF) lower in the band gap.
In this plot, Ed= 0.2. Bottom: Increasing Edwill also shift EF deeper into the gap to move the Fermi function out of the conduction band, due to fewer conduction band electrons. We plot values of Ed= [0.05, 0.2, 0.3] eV.
Chapter 3: Measurement of electronic transport properties 54
satisfies equation 3.34 to assume a lower value. Physically this accounts for the fact that fewer donors will decrease the number of electrons in the conduction band at a particular value of EF. Thus, the Fermi level must decrease to move the Fermi function out of the conduction band, and thus decrease the number of free electrons in our system.
In the bottom portion of Figure 3.4, we can see graphically the effect that changing the value of the binding energy Edof a donor electron has on the position of the Fermi level.
At a given temperature and donor concentration, increasing Ed moves the location of the Fermi level deeper into the band gap. Physically, we know that a larger binding lead to a lower fraction of ionized impurities at a given temperature; while simultaneously, the Fermi function will sweep through deeper donor states at lower values of EF, emptying them and forcing the electrons into the conduction band. These two facts together prove to us that the Fermi level must, therefore, shift to lower energies with an increase in the value of Ed. We also see that at dopant concentrations comparable to Nc, especially for deep-lying donor states such as Ed= 0.3 eV, that the Fermi level sits fairly far above the level of the donor state. This observation — necessary to account for the significant occupation of these deep states at such high concentration — will be of use to us in Chapters 5 and 6.